A new critical exponent koppa and its logarithmic counterpart koppa-hat

TL;DR

This paper introduces a new critical exponent koppa and its logarithmic counterpart koppa-hat, explaining the breakdown of hyperscaling above the upper critical dimension and extending hyperscaling validity across all dimensions.

Contribution

It identifies the distinction between correlation-length and underlying length scales above the upper critical dimension and incorporates this into a generalized hyperscaling framework.

Findings

Introduction of a new critical exponent koppa relating length scales

Derivation of logarithmic corrections at the upper critical dimension

Extension of hyperscaling to all spatial dimensions

Abstract

It is well known that standard hyperscaling breaks down above the upper critical dimension d_c, where the critical exponents take on their Landau values. Here we show that this is because, in standard formulations in the thermodynamic limit, distance is measured on the correlation-length scale. However, the correlation-length scale and the underlying length scale of the system are not the same at or above the upper critical dimension. Above d_c they are related algebraically through a new critical exponent koppa, while at d_c they differ through logarithmic corrections governed by an exponent koppa-hat. Taking proper account of these different length scales allows one to extend hyperscaling to all dimensions.